1. Exponential and Logarithmic Models (Day 2)
3.5

2. And the case of the murdered professor
Calvin (Cal) Q. Les
And the case of the murdered professor

3. Suspects LaBron James Albert Einstein
Seen arguing with Mr. Cutler at 1:30pm
Was boarding a plane heading for Cleveland, OH at Sky Harbor at 2:45pm
Seen leaving the math building at 3:05pm
Was playing at the Discovery Science Center between 10am and 2pm

4. Suspects Brad Pitt Bill Gates
Seen hanging outside Mr. Cutler’s house at 6:15pm
Was at an autograph signing from 1pm-5pm
Was part of a CHS Engineering ThinkTank group from noon to 9pm
Whereabouts unknown from 7pm to 8:15pm

5. According to Newton’s Law of Cooling, the time that elapses since death can be calculated using the model 𝑡=−10 ln 𝑇− 𝑅 𝑡 𝑁𝐵 𝑡 − 𝑅 𝑡 , where…
t is the time that has elapsed in hours
T is the Temperature of the body at a given time
Rt is the Room temperature
NBt is the Normal Body temperature

6. Just the Facts, Ma’am… Body found at 8:30pm
At 9pm, temperature of the body was 85.7℉
At 11pm, temperature of the body was 82.8℉
Room temperature was a constant 70.0℉ all weekend
Assume Mr. Cutler was healthy and had a normal body temperature of 98.6℉ at the time of death
According to Newton’s Law of Cooling, the time that elapses since death can be calculated using the model 𝑡=−10 ln 𝑇− 𝑅 𝑡 𝑁𝐵 𝑡 − 𝑅 𝑡

7. Whodunnit?

8. Objectives
Use exponential and logarithmic functions to model and solve real-life problems.

9. Example 2 – Population Growth
In a research experiment, a population of fruit flies increases according to the exponential model 𝑦=33.33 𝑒 𝑘𝑡 . After 2 days there are 100 flies. Find the value of k then determine how many flies there will be after 5 days.

10. Example 6 – Magnitudes of Earthquakes
On the Richter scale, the magnitude R of an earthquake of intensity I is given by where I0 = 1 is the minimum intensity used for comparison. Find the intensity of each earthquake. (Intensity is a measure of the wave energy of an earthquake.) a. Alaska in 2012: R = 4.0 b. Christchurch, New Zealand, in 2011: R = 6.3

11. Example 6(b) – Solution
cont’d
Note that an increase of 2.3 units on the Richter scale (from 4.0 to 6.3) represents an increase in intensity by a factor of 2,000,000/10,000 = 200. In other words, the intensity of the earthquake in Christchurch was about 200 times as great as that of the earthquake in Alaska.

12. 3.5 Example – Worked Solutions

13. Example 2 – Solution
In a research experiment, a population of fruit flies increases according to the exponential model 𝑦=33.33 𝑒 𝑘𝑡 . After 2 days there are 100 flies. Find the value of k then determine how many flies there will be after 5 days. 100=33.33 𝑒 𝑘(2) plug in known values = 𝑒 2𝑘 divide by ln =2𝑘 “Circular Rule of 3” ≈𝑘 solve for k

14. Example 2 – Solution
After 5 years 𝑦=33.33 𝑒 𝑘𝑡 yields… 𝑦=33.33 𝑒 (5) 𝑦≈520 flies

15. Example 6 – Magnitudes of Earthquakes
On the Richter scale, the magnitude R of an earthquake of intensity I is given by where I0 = 1 is the minimum intensity used for comparison. Find the intensity of each earthquake. (Intensity is a measure of the wave energy of an earthquake.) a. Alaska in 2012: R = 4.0 b. Christchurch, New Zealand, in 2011: R = 6.3

16. Example 6(a) – Solution
Because I0 = 1 and R = 4.0, you have 4.0 = = 10log I = I 10,000 = I.
Substitute 1 for I0 and 4.0 for R.
Exponentiate each side.
Inverse Property
Simplify.

17. Example 6(b) – Solution
cont’d
For R = 6.3, you have 6.3 = = 10log I = I 2,000,000 ≈ I.
Substitute 1 for I0 and 6.3 for R.
Exponentiate each side.
Inverse Property
Use a calculator.

18. Example 6(b) – Solution
cont’d
Note that an increase of 2.3 units on the Richter scale (from 4.0 to 6.3) represents an increase in intensity by a factor of 2,000,000/10,000 = 200. In other words, the intensity of the earthquake in Christchurch was about 200 times as great as that of the earthquake in Alaska.